Question

The eccentricity of the ellipse $4x^2+9y^2+8x+36y+4=0$ is

• A. $\frac{5}{6}$
• B. $\frac{3}{5}$
• C. $\frac{\sqrt{2}}{3}$
• D. $\frac{\sqrt{5}}{3}$

Answer 1

The correct option is D $\frac{\sqrt{5}}{3}$

Equation of the ellipse is

$4x^2+9y^2+8x+36y+4=0$

$\Rightarrow (2x{)}^2+8x+2^2-2^2+(3y{)}^2+36y+6^2-6^2+4=0$

$\Rightarrow (2x+2{)}^2+(3y+6{)}^2-4-36+4=0$

$\Rightarrow (2x+2{)}^2+(3y+6{)}^2=36$

$\Rightarrow 4(x+1{)}^2+9(y+2{)}^2=36$

$\Rightarrow \frac{4}{36}(x+1{)}^2+\frac{9}{36}(y+2{)}^2=1$

$\Rightarrow \frac{(x+1{)}^2}{9}+\frac{(y+2{)}^2}{4}=1$

$\Rightarrow \frac{(x+1{)}^2}{3^2}+\frac{(y+2{)}^2}{2^2}=1$

$\therefore a=3,b=2$

Eccentricity of the ellipse$=\frac{\sqrt{a^2-b^2}}{a}$

$=\frac{\sqrt{3^2-2^2}}{3}$

$=\frac{\sqrt{9-4}}{3}$

$=\frac{\sqrt{5}}{3}$.